What size screw do I need to mount / attach the ice maker to the freezer on a Kenmore elite model 596 73503200?

Go to your local hardware store and get some stainless steel metal screws that are about an inch long. The screws need to go deep enough to hold but not too long to go through the side of the freezer. No need to worry about damaging components since there aren't any in the refrigerator case in this area. The drew will just go through the inner liner and into the insulation.

I do not know if it will help, but in case you did not get the manual with it, you can find one here: http://downloadfreemanuals.com/kenmore-elite-refrigerator-manual/

Happy fixing! :)